
theorem
  for p,q,r,s being Element of absolute st
  p,q,r are_mutually_distinct & q,p,s are_mutually_distinct
  holds ex N being invertible Matrix of 3,F_Real st
  homography(N).:absolute = absolute &
  (homography(N)).p = q & (homography(N)).q = p &
  (homography(N)).r = s &
  (for t being Element of real_projective_plane st
    t in tangent p /\ tangent q holds (homography(N)).t = t)
  proof
    let p,q,r,s be Element of absolute;
    assume that
A1: p,q,r are_mutually_distinct and
A2: q,p,s are_mutually_distinct;
    consider t be Element of real_projective_plane such that
A3: tangent p /\ tangent q = {t} by A1,Th25;
A4: t in tangent p /\ tangent q by A3,TARSKI:def 1;
    then consider N1 be invertible Matrix of 3,F_Real such that
A5: homography(N1).:absolute = absolute &
    (homography(N1)).Dir101 = p &
    (homography(N1)).Dirm101 = q &
    (homography(N1)).Dir011 = r &
    (homography(N1)).Dir010 = t by A1,Th37;
    consider N2 be invertible Matrix of 3,F_Real such that
A7: homography(N2).:absolute = absolute &
     (homography(N2)).Dir101 = q &
     (homography(N2)).Dirm101 = p &
     (homography(N2)).Dir011 = s &
     (homography(N2)).Dir010 = t by A2,A4,Th37;
    reconsider N = N2 * N1~ as invertible Matrix of 3,F_Real;
A20: (homography(N)).p = (homography(N2)).((homography(N1~)).p)
                           by ANPROJ_9:13
                       .= q by A5,A7,ANPROJ_9:15;
A21: (homography(N)).q = (homography(N2)).((homography(N1~)).q)
                           by ANPROJ_9:13
                       .= p by A5,A7,ANPROJ_9:15;
A22: (homography(N)).r = (homography(N2)).((homography(N1~)).r)
                           by ANPROJ_9:13
                       .= s by A5,A7,ANPROJ_9:15;
A23: (homography(N)).t = (homography(N2)).((homography(N1~)).t)
                           by ANPROJ_9:13
                       .= t by A5,A7,ANPROJ_9:15;
    homography(N1) in EnsHomography3 by ANPROJ_9:def 1;
    then reconsider h1 = homography(N1) as Element of EnsHomography3;
    h1 is_K-isometry by A5;
    then h1 in EnsK-isometry;
    then reconsider hsg1 = h1 as Element of SubGroupK-isometry by Def05;
    homography(N2) in EnsHomography3 by ANPROJ_9:def 1;
    then reconsider h2 = homography(N2) as Element of EnsHomography3;
    h2 is_K-isometry by A7;
    then h2 in EnsK-isometry;
    then reconsider hsg2 = h2 as Element of SubGroupK-isometry by Def05;
    homography(N1~) in EnsHomography3 by ANPROJ_9:def 1;
    then reconsider h3 = homography(N1~) as Element of EnsHomography3;
A24: hsg1" = h3 by Th36;
    set H = EnsK-isometry,
        G = GroupHomography3;
    reconsider hg1 = hsg1, hg2 = hsg2, hg3 = hsg1" as Element of G
      by A24,ANPROJ_9:def 4;
    reconsider hsg3 = h3 as Element of SubGroupK-isometry by A24;
    reconsider h4 = hsg2 * hsg3 as Element of SubGroupK-isometry;
A25: h4 = hg2 * hg3 by A24,GROUP_2:43
       .= h2 (*) h3 by A24,ANPROJ_9:def 3,def 4
       .= homography N by ANPROJ_9:18;
    h4 in the carrier of SubGroupK-isometry;
    then h4 in EnsK-isometry by Def05;
    then consider h be Element of EnsHomography3 such that
A26: h4 = h and
A27: h is_K-isometry;
    take N;
    thus homography(N).:absolute = absolute by A25,A26,A27;
    thus (homography(N)).p = q & (homography(N)).q = p &
      (homography(N)).r = s by A20,A21,A22;
    thus for t being Element of real_projective_plane st
      t in tangent p /\ tangent q holds (homography(N)).t = t
    proof
      let v be Element of real_projective_plane;
      assume v in tangent p /\ tangent q;
      then v = t by A3,TARSKI:def 1;
      hence thesis by A23;
    end;
  end;
